Geodesic
Justification of the wavelet-based integral representation of a solution of the wave equation
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 107-123 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study the previously obtained integral representation of a solution of the wave equation. The integrand contains the weighted localized solutions of the wave equation, which depend on integration variables. We construct the parameterized family of the localized solutions from the chosen one by employing the transformations of shift, dilation and the Lorentz transform. We give the sufficient conditions of the pointwise convergence for the obtained improper integral. The convergence in the $\mathcal L_2$ sense is proven too. 
@article{ZNSL_2017_461_a5,
     author = {E. A. Gorodnitskiy and M. V. Perel},
     title = {Justification of the wavelet-based integral representation of a~solution of the wave equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {107--123},
     year = {2017},
     volume = {461},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a5/}
}
TY  - JOUR
AU  - E. A. Gorodnitskiy
AU  - M. V. Perel
TI  - Justification of the wavelet-based integral representation of a solution of the wave equation
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 107
EP  - 123
VL  - 461
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a5/
LA  - ru
ID  - ZNSL_2017_461_a5
ER  - 
%0 Journal Article
%A E. A. Gorodnitskiy
%A M. V. Perel
%T Justification of the wavelet-based integral representation of a solution of the wave equation
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 107-123
%V 461
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a5/
%G ru
%F ZNSL_2017_461_a5
E. A. Gorodnitskiy; M. V. Perel. Justification of the wavelet-based integral representation of a solution of the wave equation. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 107-123. http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a5/

[1] M. V. Perel, “Integral representation of solutions of the wave equation based on Poincaré wavelets”, Proceedings of the International Conference Days on Diffraction, Saint-Petersburg, 2009, 159–161

[2] E. Gorodnitskiy, M. V. Perel, “The Poincaré wavelet transform: implementation and interpretation”, Proceedings of the International Conference Days on Diffraction, Saint-Petersburg, 2011, 72–77

[3] M. V. Perel, E. Gorodnitskiy, “Integral representations of solutions of the wave equation based on relativistic wavelets”, J. Phys. A: Math. Theor., 45:38 (2012), 385203 http://stacks.iop.org/1751-8121/45/i=38/a=385203 | DOI | MR | Zbl

[4] G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser, 1994 | MR | Zbl

[5] M. V. Perel, M. S. Sidorenko, “Wavelet analysis for the solutions of the wave equation”, Proceedings of the International Conference Days on Diffraction, Saint-Petersburg, 2006, 208–217 | DOI

[6] M. V. Perel, M. S. Sidorenko, “New physical wavelet 'Gaussian wave packet' ”, J. Phys. A: Math. Theor., 40:13 (2007), 3441 http://stacks.iop.org/1751-8121/40/i=13/a=011 | DOI | MR | Zbl

[7] M. V. Perel, M. S. Sidorenko, “Wavelet-based integral representation for solutions of the wave equation”, J. Phys. A: Math. Theor., 42:37 (2009), 375211 http://stacks.iop.org/1751-8121/42/i=37/a=375211 | DOI | MR | Zbl

[8] A. P. Kiselev, M. V. Perel, “Highly localized solutions of the wave equation”, J. Math. Phys., 41:4 (2000), 1034–1955 | DOI | MR

[9] A. P. Kiselev, M. V. Perel, “Gaussovskie volnovye pakety”, Optika i Spektroskopiya, 86:3 (1999), 357–359

[10] E. Gorodnitskiy, M. Perel, Yu Geng, R.-S. Wu, “Depth migration with Gaussian wave packets based on Poincaré wavelets”, Geophys. J. International, 205:1 (2016), 314–331 | DOI

[11] V. M. Babich, Yu. P. Danilov, “Postroenie asimptotiki resheniya uravneniya Shredingera, sosredotochennoi v okrestnosti klassicheskoi traektorii”, Zap. nauchn. sem. LOMI, 15, 1969, 47–65 | MR | Zbl

[12] V. M. Babich, V. V. Ulin, “Kompleksnyi prostranstvenno-vremennoi luchevoi metod i ‘kvazifotony’ ”, Zap. nauchn. sem. LOMI, 117, 1981, 5–12 | MR | Zbl

[13] A. P. Kachalov, “Sistema koordinat pri opisanii ‘kvazifotona’ ”, Zap. nauchn. sem. LOMI, 140, 1984, 73–76 | MR | Zbl

[14] V. M. Babich, “Kvazifotony i prostranstvenno-vremennoi luchevoi metod”, Zap. nauchn. sem. POMI, 342, 2007, 5–13 | MR | Zbl

[15] M. M. Popov, “Novyi metod rascheta volnovykh polei v vysokochastotnom priblizhenii”, Zap. nauchn. sem. LOMI, 104, 1981, 195–216 | MR | Zbl

[16] M. Leibovich, E. Heyman, “Beam Summation Theory for Waves in Fluctuating Media. Part I: The Beam Frame and the Beam-Domain Scattering Matrix”, IEEE Transactions on Antennas and Propagation, 65:10 (2017), 5431–5442 | DOI | MR

[17] M. Leibovich, E. Heyman, “Beam Summation Theory for Waves in Fluctuating Media. Part II: Stochastic Field Representation”, IEEE Transactions on Antennas and Propagation, 65:10 (2017), 5443–5452 | DOI | MR

[18] A. N. Norris, “Complex point-source representation of real point sources and the Gaussian beam summation method”, J. Opt. Soc. Am. A, 3:12 (1986), 2005–2010 | DOI

[19] S. T. Ali, J.-P. Antoine, J. P. Gazeau, Coherent States, Wavelets, and Their Generalizations, Springer-Verlag, New York, 1999 | MR

[20] A. Grossmann, J. Morlet, T. Paul, “Transforms associated to square integrable group representations. I. General results”, J. Math. Phys., 26:10 (1985), 2473–2479 | DOI | MR | Zbl

[21] J.-P. Antoine, R. Murenzi, P. Vandergheynst, S. T. Ali, Two-Dimensional Wavelets and their Relatives, Cambridge University Press, 2004 | MR | Zbl

[22] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992 | MR | Zbl